Feeble fish in time-dependent waters and homogenization of the G-equation
aa r X i v : . [ m a t h . O C ] A p r FEEBLE FISH IN TIME-DEPENDENT WATERSAND HOMOGENIZATION OF THE G-EQUATION
DMITRI BURAGO, SERGEI IVANOV, AND ALEXEI NOVIKOV
Abstract.
We study the following control problem. A fish with bounded aquatic loco-motion speed swims in fast waters. Can this fish, under reasonable assumptions, get toa desired destination? It can, even if the flow is time-dependent. Moreover, given a pre-scribed sufficiently large time t , it can be there at exactly the time t . The major differencefrom our previous work is the time-dependence of the flow. We also give an application tohomogenization of the G-equation. Introduction
Let V = V t be a time-dependent vector field in R n , n ≥
2. We assume that V t ( x ) iscontinuous, uniformly bounded, and locally Lipschitz in x . We often abuse the language andrefer to V t as a flow . Definition 1.1.
An absolutely continuous path γ : [ t , t ] → R n is said to be admissible if (cid:12)(cid:12)(cid:12)(cid:12) ddt γ ( t ) − V t ( γ ( t )) (cid:12)(cid:12)(cid:12)(cid:12) ≤ t ∈ [ t , t ].Let x , x ∈ R n , t , t ∈ R , t ≤ t . We say that a point ( x , t ) in space-time is reachablefrom ( x , t ) if there exists an admissible path γ : [ t , t ] → R n with γ ( t ) = x and γ ( t ) = x .If ( x , t ) is reachable from ( x , t ), we also say that x is reachable from ( x , t ) at time t . In the sequel we usually assume that the initial conditions are x = 0 and t = 0. Forbrevity, we say that x is reachable at time t if ( x, t ) is reachable from (0 , V t is thevelocity field of waters in an ocean. Fish living in the ocean have bounded aquatic locomotivespeed. We normalize the data so that the maximal speed of the fish is 1, and the speed ofwaters can be much larger. Definition 1.1 formalizes the condition that a fish starting itsjourney from x at time t can control its motion so that it finds itself at x at exactlytime t .A similar problem was considered in [5, 11] for time-independent vector fields V and aweaker reachability result: the fish is not required to arrive at its destination exactly at aprescribed time. Mathematics Subject Classification.
Key words and phrases.
G-equation, small controls, time-dependant incompressible flow, reachability.The first author was partially supported by NSF grant DMS-1205597. The second author was partiallysupported by RFBR grant 17-01-00128. The third author was partially supported by NSF grants DMS-1515187 and DMS-1813943. andling time-dependence of V t required considerable effort and actually forced us to provea stronger result. This reachability problem is directly related to the G-equation which inparticular models combustion processes in the presence of turbulence. Therefore anothersubstantial part of this paper is an application to homogenization of the G-equation. Weaddress this application in Section 6.Our main result, see Theorem 1.2 below, states that under natural assumptions on V t everypoint is reachable at all sufficiently large times. The assumptions on V t are the following:(i) The field V t ( x ) is bounded: M := 1 + sup t,x | V t ( x ) | < ∞ , and is locally Lipschitz in x .(ii) The flow is incompressible: div V t = 0 for all t.(iii) Small mean drift:(1.1) lim L →∞ sup t ∈ R ,x ∈ R n (cid:13)(cid:13)(cid:13)(cid:13) L n Z [0 ,L ] n V t ( x + y ) dy (cid:13)(cid:13)(cid:13)(cid:13) = 0 . All assumptions (i)-(iii) are essential. First, the flow might have a sink towards which theflow runs faster than the maximum possible speed the fish can swim. This issue is easilyresolved by the assumption (ii) that the flow is incompressible. Next, the velocity of the flowmight point in one direction and again it may have speed greater than the maximal speed ofthe fish. This obstruction is resolved by the condition (iii) of small mean drift on the largescale. Finally, the flow could be so strong that the fish is carried to infinity in finite time.The condition (i) rules out this possibility. The condition (i) is also a technical assumptionwhich is needed to be able to formulate the problem formally.It was a surprise to us that, under these modest assumptions the fish can reach everydestination point x ∈ R n . Furthermore, there is some t x such that if t ≥ t x , the fish can getto x at exactly time t . We also prove an asymptotically optimal bounds for the reach time,namely t x grows no faster than | x | as | x | → ∞ .Now we are in a position to formulate our main result. Theorem 1.2.
For every flow V t satisfying (i)–(iii) above and every a > , there exists C > such that for all x , x ∈ R n and t ∈ R , ( x, t ) is reachable from ( x , t ) for every t ≥ t + a | x − x | + C .Remark. The constant C in Theorem 1.2 depends on a and parameters of the flow. One cancheck that C can be determined in terms of a , the parameter M from (i), and the rate ofconvergence of the mean drift to zero in (iii).The small mean drift assumption (iii) may be relaxed at the expense of a weaker estimateon the reach time. Namely we have the following. Corollary 1.3.
Let V t be a flow satisfying (i), (ii), and (1.2) ∆ := inf L> sup t ∈ R ,x ∈ R n (cid:13)(cid:13)(cid:13)(cid:13) L n Z [0 ,L ] n V t ( x + y ) dy (cid:13)(cid:13)(cid:13)(cid:13) < . Then for every a > − ∆ there exists C > such that for all x , x ∈ R n and t ∈ R , ( x, t ) isreachable from ( x , t ) for every t ≥ t + a | x − x | + C . he gist of the proof of Theorem 1.2 is: Fix a flow V t and assume without loss of generalitythat x = 0 and t = 0. For t, r > R t denote the set of points reachable at time t and I r the cube [ − r, r ] n in R n . Our goal is to show that, for every fixed r and for all sufficientlylarge t the set R t contains I r . We do this analyzing the volume of the intersection R t ∩ I r as a function of t .The paper is organized as follows. In Section 2 we introduce our notation and maintools. In particular, there we discuss isoperimetric inequalities, co-area formula, slicing, andcertain regularity results such as rectifiability of the boundary of the reachable set. Severalimportant facts about BV-functions can be found in Appendix A. In Section 3 we proveTheorem 1.2 and Corollary 1.3. Sections 4 and 5 provide auxiliary estimates needed in theproof of Theorem 1.2. In Section 6 we give an application of Theorem 1.2 to the theory ofrandom homogenization of the G-equation. Some further directions.
In a discussion with the first author, Leonid Polterovich sug-gested to consider a similar problem where the fish is not a point but rather a region (thinkof an amoeba or a jelly-fish, for instance). Leonid suggested the following symplectic formu-lation. Let us say we are in R and the flow is Hamiltonian. This, of course, means that thearea of the fish does not change but its shape may change. The fish has a fixed amount ofHofer’s energy it can spend to change the flow. In two dimensions Hofer’s energy is E ( u ) = Z ∞−∞ [ sup x ∈ R ( ψ ( x, t )) − inf x ∈ R ( ψ ( x, t ))] dt, where ψ ( x, t ) is the stream-function (Hamiltonian) of the flow u ( x, t ). Now the problem inquestion is as follows: Initially the fish sits in some ball, and it wants to get to another(destination ball) of the same size. Leonid has made the following observation, which firstsounds very counter-intuitive. If the flow is constant (possibly very fast, no small mean drift),the fish can get from any ball to a ball of the same size located in the direction opposite tothe flow and very far. Using the same amount of Hofer’s energy, the fish can swim againstan arbitrarily fast flow arbitrarily far away!We do not include a formal proof here. Here is an intuitive description. Assume that thefish has M worth of Hofer’s energy, where M depends on the radius of the initial ball. Itspends M/ M/ M/ he first naive idea that comes to one’s mind is to impose restrictions on the potential energyof the membrane (to keep the amoeba in one piece, at least) and on kinetic energy (for it isstill “feeble”). We have not made any progress in this direction so far.2. Notation and preliminaries
Let I r = [ − r, r ] n denote the cube with edge length 2 r centered at 0, B r ( x ) the Euclideanball of radius r centered at x ∈ R n , and V n = | B (0) | the volume of the unit ball in R n .Occasionally we use r = ∞ , with the convention that I ∞ = B ∞ ( x ) = R n .For x ∈ R n , t ∈ R and t ≥
0, we denote by R t ( x , t ) the set of points reachable from( x , t ) at time t + t , see Definition 1.1. For brevity, let R t = R t (0 , R t ∩ I r is denoted by w ( r, t ):(2.1) w ( r, t ) = |R t ∩ I r | = Z I r χ R t ( x ) dx, where χ R t is the characteristic function of the reachable set R t . The volume w ( r, t ) is themain quantity of interest.Recall that the maximum control in Definition 1.1 is bounded by 1. Hence | x − x | ≤ M t if x is reachable from ( x , t ) at time t + t , where M is defined in the condition (i) above.Therefore(2.2) R t ⊂ B tM (0) ⊂ I tM for all t >
0. Hence R t ∩ I r = R t if r ≥ tM .We now define s ( r, t ) ≥
0, the perimeter of R t inside the cube I r . As we discuss below, s ( r, t ) is essentially the ( n − ∂ R t ∩ I r . Theformal definition is based on the notion of total variation for BV functions, see Appendix A,in particular Definition A.2. Namely s ( r, t ) := P ( R t , I ◦ r ) = Var( χ R t , I ◦ r ) , where I ◦ r is the interior of I r . Here the last expression is the variation of the characteristicfunction χ R t in I ◦ r , see Definition A.1.Denote D r ( t ) := R t ∩ ∂I r . The following lemma estimates the rate of change of the volume of R t . It is the maintechnical tool in our proof. Lemma 2.1.
For any fixed r > , (2.3) ddt w ( r, t ) ≥ s ( r, t ) − flux( V t , D r ( t )) in the sense of distributions (with respect to t ), where flux( V t , D r ( t )) is the flux of the vectorfield V t through the ( n − -dimensional “surface” D r ( t ) ⊂ ∂I r . Formally flux( V t , D r ( t )) isdefined by flux( V t , D r ( t )) = Z D r ( t ) V t ( x ) · ν ( x ) dx where ν ( x ) is the outer normal to the boundary of the cube I r at a point x ∈ ∂I r . n the case r = ∞ we also have (2.3) , in the form (2.4) ddt w ( ∞ , t ) ≥ s ( ∞ , t ) . Remark.
The inequalities (2.3) and (2.4) are easy to verify in the case when V t is smoothand the boundary of R t is a smooth hypersurface transverse to ∂I r . In fact, in this casethe inequalities turn into equalities. Indeed, for a small δ > R t to R t + δ is approximately the composition of two operations: First move the reachable set time δ along the flow and then replace the resulting set by its δ -neighborhood. The first operationdoes not change the volume of the set since the flow is incompressible. However, the volumeof the intersection with I r changes, it is reduced by the amount of the flow that leaks outthrough the boundary of I r . This amount is approximately δ · flux( V t , D r ( t )). On the secondstep, taking the δ -neighborhood increases the volume by approximately δ · s ( r, t ), since s ( r, t )is the area of the relevant part of the boundary of R t . Passing to the limit as δ → R t has arectifiable topological boundary (compare with [5, § Proof of Lemma 2.1.
The relation (2.4) follows from (2.3) and (2.2). To prove (2.3), considera family of functions u ε : R n × R + → R , ε >
0, defined by(2.5) u ε ( x, t ) = sup { e −| y | /ε | y ∈ R n is such that x ∈ R t ( y, } . Equivalently, one can set u ε ( x ) = e −| x | /ε for all x ∈ R n and define(2.6) u ε ( x, t ) = sup { u ε ( γ (0)) | γ : [0 , t ] → R n is an admissible path with γ ( t ) = x } , see Definition 1.1. We need two properties of u ε : For every fixed ε >
0, the function u ε islocally Lipschitz and it satisfies the following partial differential equation:(2.7) ∂ t u ε + V t · ∇ u ε = |∇ u ε | for a.e. x ∈ R n and t >
0, where ∇ u ε denotes the gradient of u ε with respect to the firstargument. The equation (2.7) is called the G-equation associated to V t .The above properties are not hard to verify directly. Alternatively, one can prove themusing the theory of viscosity solutions, as follows. The equation (2.7) is a Hamilton-Jacobiequation with the Hamiltonian H ( t, x, p ) = −| p | + V t · p and the corresponding Lagrangian L ( t, x, q ) = inf p ∈ R n [ p · q − H ( t, x, p )] = ( , if | q − V t | ≤ , −∞ , otherwise.By e.g. [7, Theorem 7.2], the function u ε defined by (2.6) is a viscosity solution of (2.7) withthe initial data u ε ( x,
0) = u ε . For a definition, motivations, and derivation of viscosity solu-tions for optimal control problems see e.g. [2]. Since u ε is bounded and uniformly continuousand V t is locally Lipschitz and bounded, the viscosity solution u ε ( x, t ) is locally Lipschitz(by e.g. Lemma 9.2 in [4]). Furthermore, a viscosity solution satisfies the equation whenever t is differentiable (see e.g. Proposition 1.9 on p.31 in [2]). Hence by Rademacher’s Theorem u ε satisfies (2.7) almost everywhere.The formula (2.5) implies that u ε ( x, t ) ↓ χ R t ( x ) as ε ↓
0, where χ R t is the characteristicfunction of R t . Hence Z I r u ε ( x, t ) dx → w ( r, t )and flux( V t u ε , ∂I r ) → flux( V t , D r ( t ))as ε →
0. Integrating the G-equation over I r and taking into account the incompressibilityof V t we obtain that ∂ t Z I r u ε dx + flux( V t u ε , ∂I r ) = Z I r |∇ u ε | dx. Hence for any t and t we have Z t t Var( u ε , I ◦ r ) dt = Z t t Z I r |∇ u ε | dxdt = Z I r u ε ( x, t ) dx − Z I r u ε ( x, t ) dx + Z t t flux( V t u ε , ∂I r ) dt. Note that this quantity is bounded by a constant independent of ε since | u ε | ≤ | V t | ≤ M . By Fatou’s Lemma and the lower semi-continuity of the total variation (see e.g.Remark 3.5 in [1]) it follows that Z t t s ( r, t ) dt ≡ Z t t Var( χ R t , I ◦ r ) dt ≤ lim inf ε → Z t t Var( u ε , I ◦ r ) dxdt. Thus Z t t s ( r, t ) dt ≤ w ( r, t ) − w ( r, t ) + Z t t flux( V t , D r ( t )) dt. This inequality means that (2.3) holds in the sense of distributions. (cid:3)
Remark.
Since flux( V t , D r ( t )) is bounded for every fixed r and s ( r, t ) ≥
0, Lemma 2.1 impliesthat w ( r, · ) is the sum of a Lipschitz function and a non-decreasing function. Therefore foralmost all t > ddt w ( r, t ) exists and satisfies (2.3).By (2.4) the perimeter P ( R t ) = s ( ∞ , t ) is finite for almost all t >
0. This and theDe Giorgi Theorem A.4 imply that the perimeter of R t equals the ( n − H n − ( ∂ ∗ R t ) of a rectifiable set ∂ ∗ R t , the reduced boundary of R t (seeDefinition A.3). We define p ( r, t ) to be the ( n − ∂ ∗ R t by ∂I r :(2.8) p ( r, t ) = H n − ( ∂ ∗ R t ∩ ∂I r ) . Then Corollary A.8 gives us the co-area inequality for this slicing:(2.9) s ( r , t ) − s ( r , t ) ≥ Z r r p ( x, t ) dx. The quantity p ( r, t ) can be though of as the ( n − n − D r ( t ) = R t ∩ ∂I r . This is formalized in the appendix (see Theorem A.9) andused in the proof of Lemma 4.4 below.We will need the following isoperimetric inequalities. he Euclidean Isoperimetric Inequality (Theorem 14.1 in [12]) implies that the volume w ( ∞ , t ) = |R t | of the entire reachable set R t and its perimeter s ( ∞ , t ) satisfy(2.10) s ( ∞ , t ) ≥ λ w ( ∞ , t ) n − n , where λ = n V /nn is the Euclidean isoperimetric constant satisfying | ∂B r (0) | = λ | B r (0) | n − n for all r > . The Relative Isoperimetric Inequality in the cube (Theorem A.5 in Appendix) implies thatthe volume w ( r, t ) of R t ∩ I r and its relative perimeter s ( r, t ) inside I r satisfy(2.11) s ( r, t ) ≥ λ (cid:0) min { w ( r, t ) , | I r | − w ( r, t ) } (cid:1) n − n , where λ is a positive constant depending only on n .3. Proof of Theorem 1.2 and Corollary 1.3
Most of this section we spend proving Theorem 1.2. Its most technical stage (namely theproof of Proposition 3.2) is put off. It is contained in Sections 4 and 5.Let us say a few words about how the proof of Theorem 1.2 goes. It is easy to show thatthe volume of R t grows to infinity. It is a more delicate task to verify that the set R t cannotbe carried away from the origin by the flow. Our idea is to show that, for every r ≥
0, the set R t ∩ I r fills I r for all sufficiently large t . Thus we look at how the volume w ( r, t ) = |R t ∩ I r | grows. We want it to reach (2 r ) n , the volume of I r . This is done by dividing the fillingprocess into three stages. During the initial stage we fill in at least α | I r | of the volume of I r , where α is a small positive constant defined below. In the next step, which is the keyone, we fill in at least (1 − α ) | I r | of the volume of I r . Furthermore, this portion of volumeremains filled forever after a certain time t . Finally, we show that at a later time a smallercube I r/ is completely filled. Since the choice of r is arbitrary, r/ r .Our choice of α depends on the maximal speed of the fluid flow and the dimension. Wefix(3.1) α = V n (4 M ) n for the rest of the proof. We assume that r is sufficiently large, more precisely r ≥ r where r is a constant depending on V t . The precise value of r is defined in the course of the proof.The initial stage of the filling process is simple. It is analyzed in the following lemma: Lemma 3.1.
Let r > and T = r M . Then w ( r, T ) ≥ α | I r | . Proof.
By (2.2) we have R T ⊂ I r , hence w ( r, T ) = |R T | . Clearly R t has a nonemptyinterior and hence |R t | > t >
0. By (2.4) and the isoperimetric inequality (2.10)we have ddt |R t | ≥ s ( ∞ , t ) ≥ n V /nn |R t | n − n . Therefore(3.2) |R t | ≥ V n · t n = | B t (0) | . Hence w ( r, T ) ≥ V n T n = V n (2 M ) − n r n = α | I r | . (cid:3) he middle stage of the filling process is the most technical. This is the content of thenext proposition. Proposition 3.2.
There exist constants A = A ( n ) ≥ and r > such that w ( r, t ) > (1 − α ) | I r | for all r ≥ r and t ≥ Ar . We prove Proposition 3.2 in Section 5. For this proof we need to estimate how muchvolume of R t ∩ I r can leak out through the boundary of I r . This estimate is contained inSection 4, see Proposition 4.1.The final stage of the filling process is simple again. It is analyzed in Lemma 3.3. We showthat, once w ( r, t ) exceeds (1 − α ) | I r | , then in time T the reachable set covers the smallercube I r/ . Lemma 3.3.
Suppose that r > and t > are such that w ( r, t ) > (1 − α ) | I r | . As in theprevious lemma, let T = r M . Then I r/ ⊂ R t + T . Proof.
Fix p ∈ I r/ and let t = t + T . Let R − t = { x ∈ R n : ( p, t ) is reachable from ( x, t − t ) } . R − t is the reachable set from p for the reversed flow V − t = − V t − t . As in the previous lemmawe can apply (3.2) to V − t to obtain |R − T | ≥ V n T n = α | I r | . Hence |R − T | + w ( r, t ) > | I r | . By (2.2) applied to V − t we have R − T ⊂ B r/ ( p ) ⊂ I r . Thus R − T ∩ R t = ∅ . Hence p ∈ R t . (cid:3) Combining the results of the three stages, we obtain the following proposition, which isessentially Theorem 1.2 with non-optimal bounds on reach time.
Proposition 3.4.
There exist constants µ = µ ( n ) ∈ (0 , and C > such that for every t ≥ C we have B µt (0) ⊂ R t .Proof. By Proposition 3.2 we have w ( r, t ) > (1 − α ) | I r | for all r ≥ r and t ≥ Ar . By Lemma3.3 it follows that B r/ (0) ⊂ I r/ ⊂ R t for all t ≥ Ar + T = ( A + M ) r . Applying this to 2 r in place of r yields that B r (0) ⊂ R t for all r ≥ r and t ≥ (2 A + 1) r . Hence the statement holds for µ = (2 A + 1) − and C = (2 A + 1) r . (cid:3) Now we are in a position to prove Theorem 1.2 and Corollary 1.3.
Proof of Theorem 1.2.
Fix ε >
0. Note that Proposition 3.4 (after a suitable rescaling) holdsfor controls bounded by ε instead of 1. Our plan is to spare a small part of control to ensurereachability and use the remaining part of control to add the drift with speed 1 − ε in adesired direction. ithout loss of generality assume that x = 0 and t = 0. Fix v ∈ R n such that | v | ≤ − ε and apply Proposition 3.4 to the flow e V defined by e V t ( x ) = 1 ε V t ( εx + tv ) . This yields a constant C ε,v > t ≥ C ε,v the reachable set for e V attime t contains the ball B µt (0). Here µ = µ ( n ) is the constant from Proposition 3.4. If e γ : [0 , t ] → R n is an admissible path for e V , then the path γ defined by γ ( τ ) = ε e γ ( τ ) + τ v is admissible for our flow V . Hence the reachable set R t contains the ball B εµt ( tv ). Inparticular the point y = tv can be reached at time t , which satisfies t ≤ | y | / (1 − ε ).It remains to show that the constant C ε,v can be chosen independently of v . To show this,let us choose a finite εµ -net { v , . . . , v m } in the ball B − ε (0) and let C ε = max { C ε,v i : 1 ≤ i ≤ m } . Then for every t ≥ C ε we have R t ⊃ m [ i =1 B εµt ( tv i ) ⊃ B (1 − ε ) t (0) . Thus every point x ∈ R n is reachable at any moment t ≥ max { C ε , | x | / (1 − ε ) } . To finishthe proof of the theorem, set ε = 1 − a and C = C ε . (cid:3) Proof of Corollary 1.3.
The idea is to use a part of the control to compensate the mean driftat some scale. Fix c ∈ (∆ , L > V t ( x ) := 1 L n Z [0 ,L ] n V t ( x + y ) dy satisfies k V t ( x ) k < c for all t and x . Let V t = V t − V t . Then(3.4) (cid:13)(cid:13)(cid:13)(cid:13) L n Z [0 ,L ] n V t ( x + y ) dy (cid:13)(cid:13)(cid:13)(cid:13) ≤ n L ML for all L ≥ L . Indeed, let Φ L denote the characteristic function of the cube [ − L, n divided by L n . Then V t is the convolution V t ∗ Φ L and the integral in (3.4) is the valueat x of the convolution V t ∗ Φ L = V t ∗ (Φ L − Φ L ∗ Φ L ). The function | Φ L − Φ L ∗ Φ L | isbounded by 1 /L n and its support is contained in the set [ − L − L , n \ [ − L, − L ] n of volume( L + L ) n − ( L − L ) n ≤ n L L n − . Hence the L -norm of Φ L − Φ L ∗ Φ L is bounded by2 n L /L and (3.4) follows.Observe that V t is incompressible and bounded by M . This and (3.4) imply that V t satisfies the assumptions of Theorem 1.2. We apply Theorem 1.2 to V t with the maximalfish speed set to 1 − c instead of 1. Since k V t k < c , every admissible path in this settingis admissible for the original flow V t = V t + V t . Because of the speed renormalization, theconclusion of the theorem holds for any a > − c . Since c ∈ (∆ ,
1) is arbitrary, Corollary 1.3follows. (cid:3)
Remark 3.5.
One can see from the proof that the constant C in Corollary 1.3 is determinedby M , ∆, a , and any value L such that V t ( x ) in (3.3) is bounded by ∆+12 for all t and x . . Volume change estimate
Throughout the paper we integrate areas and perimeters over time intervals. Such integralsare indicated by a hat. Namely we define b s ( r, t, T ) = Z t + Tt s ( r, τ ) dτ and b p ( r, t, T ) = Z t + Tt p ( r, τ ) dτ. The goal of this section is to prove the following proposition.
Proposition 4.1.
For every ε > there exists r > such that for all r ≥ r , t > , T ∈ [0 , r ] we have (4.1) w ( r, t + T ) − w ( r, t ) ≥ b s ( r, t, T ) − εr n . For the proof of Proposition 4.1 we need the following two lemmas.
Lemma 4.2.
For all r, t, T > , (4.2) b s ( r, t, T ) ≤ C ( r + T ) r n − where C = n n M .Proof. From Lemma 2.1 and a trivial estimate | flux( V t , D r ( t )) | ≤ M | ∂I r | we have ddt w ( r, t ) ≥ s ( r, t ) − M | ∂I r | (in the sense of distributions). By integrating this we obtain w ( r, t + T ) − w ( r, t ) ≥ b s ( r, t, T ) − M T | ∂I r | . The left-hand side is bounded above by | I r | . Hence b s ( r, t, T ) ≤ | I r | + M T | ∂I r | . Since | I r | = 2 n r n and | ∂I r | = n n r n − , (4.2) follows. (cid:3) The incompressibility and small mean drift assumptions imply the following lemma, whichwe borrow from [5]. This is the only place in the proof where the small mean drift assumptionis used.
Lemma 4.3 (cf. [5, Lemma 3.1]) . For every ε > there exists L > such that the followingholds. Let F be an ( n − -dimensional cube with edge length L ≥ L , then (4.3) | flux( V, F ) | ≤ εL n − . Proof.
This lemma is stated in [5] for a time-independent vector field. We apply [5, Lemma3.1] to the vector field V t for every fixed t . The constant L (named A in [5, Lemma 3.1])depends on the vector field, so we need to make sure that it can be chosen independently of t . In the proof in [5] one can see that L depends only on M and on the rate of convergenceof the mean drift to 0. Hence the proof works for our Lemma 4.3 as well. (cid:3) emma 4.4 (cf. [5, Lemma 3.3]) . For every ε > there exist r > and C > such thatfor almost all t > and r ≥ r , (4.4) | flux( V t , D r ( t )) | ≤ C p ( r, t ) + εr n − . Proof.
This lemma could also be borrowed from [5] if we had proven certain regularityproperties of R t . For the sake of completeness we include a proof here. The proof isessentially the same but it is based on different foundations in Geometric Measure Theory.We fix ε > L be the constant provided by Lemma 4.3. Let t > R t has finite perimeter. Assume that r ≥ L and the following holds:For every hyperplane Σ containing one of the ( n − I r , theslice R t ∩ Σ has finite perimeter in Σ ∼ = R n − and its reduced boundary in Σ coincides withΣ ∩ ∂ ∗ R t up to a set of zero ( n − r .Since r ≥ L , we have r = mL for some L ∈ [ L , L ] and m ∈ Z . We divide ∂I r into( n − F i , i = 1 , , . . . , nm n − , with edge length L . Denote D = D r ( t )for brevity. For each i , define s i = min {| F i ∩ R t | , | F i \ R t |} = min {| F i ∩ D | , | F i \ D |} and p i = P n − ( D, F ◦ i ) = H n − ( F ◦ i ∩ ∂ ∗ R t )where P n − denotes the perimeter in the respective hyperplane and F ◦ i is the relative interiorof F i . The last identity follows from the De Giorgi Theorem A.4.The isoperimetric inequality in ( n − s i ≤ CLp i where C is a constant depending only on n . For n ≥
3, we prove this isoperimetric inequalityin Appendix, Corollary A.6. For n = 2 Corollary A.6 is trivially true. Therefore, we have (cid:12)(cid:12) | flux( V t , F i ∩ D ) | − | flux( V t , F i \ D ) | (cid:12)(cid:12) ≤ | flux( V t , F i ) | ≤ εL n − , where the second inequality follows from Lemma 4.3. At least one of the quantities | flux( V t , F i ∩ D ) | and | flux( V t , F i \ D ) | is bounded by M s i , hence both of them are bounded by M s i + εL n − .Thus | flux( V t , F i ∩ D ) | ≤ M s i + εL n − ≤ CM Lp i + εL n − ≤ C p i + ε | F i | , where C = 2 CM L . Summing up over all i yields that | flux( V t , D ) | ≤ C X p i + ε | ∂I r | ≤ C p ( r, t ) + n n εr n − for almost all r ≥ L . Since ε is arbitrary, the lemma follows. (cid:3) Proof of Proposition 4.1.
Fix β, ε >
0. We apply Lemma 4.4 to ε := ε/ n +1 in place of ε .This yields | flux( V t , D r ( t )) | ≤ C p ( r, t ) + ε r n − for almost all r ≥ r and t >
0. This and (2.3) imply ddt w ( r, t ) ≥ s ( r, t ) − C p ( r, t ) − ε r n − for almost all r > r and t >
0. Integration in t yields(4.5) w ( r, t + T ) − w ( r, t ) ≥ b s ( r, t, T ) − C b p ( r, t, T ) − ε T r n − or almost all r > r and all t, T > h := 2 n +1 C C ε , where C is the constant from Lemma 4.2. By the co-area inequality (2.9), s ( r + h, t ) ≥ Z r + h p ( x, t ) dx ≥ Z r + hr p ( x, t ) dx for all r > t >
0. Once again, integration in t yields(4.6) b s ( r + h, t, T ) ≥ Z r + hr b p ( x, t, T ) dx for all r > t, T > r and t be as in the formulation of Proposition 4.1. Namely t > r ≥ r where r is to be chosen later, and 0 ≤ T ≤ r . We require that r ≥ r and r ≥ h ,the latter ensures that h ≤ r . By Lemma 4.2 applied to r + h in place of r , b s ( r + h, t, T ) ≤ C ( r + h + T )( r + h ) n − ≤ n +1 C r n since T ≤ r and h ≤ r . This and (4.6) imply that there exists ˜ r ∈ [ r, r + h ] such that(4.7) b p (˜ r, t, T ) ≤ n +1 C r n h = C − ε r n , where the equality follows from the definition of h . Furthermore the set of ˜ r ∈ [ r, r + h ]satisfying (4.7) has positive measure, hence we can choose ˜ r so that (4.7) holds and (4.5)applies to ˜ r in place of r : w (˜ r, t + T ) − w (˜ r, t ) ≥ b s (˜ r, t, T ) − C b p (˜ r, t, T ) − ε T ˜ r n − . This estimate, (4.7), and the inequalities T ≤ r and ˜ r ≤ r imply that w (˜ r, t + T ) − w (˜ r, t ) ≥ b s (˜ r, t, T ) − ε r n − n − ε r n ≥ b s (˜ r, t, T ) − n ε r n = b s (˜ r, t, T ) − εr n . Since ˜ r ≥ r , we have b s (˜ r, t, T ) ≥ b s ( r, t, T ). Thus(4.8) w (˜ r, t + T ) − w (˜ r, t ) ≥ b s ( r, t, T ) − εr n . Now we estimate the difference between w (˜ r, t + T ) and w ( r, t + T ): w (˜ r, t + T ) − w ( r, t + T ) = |R t + T ∩ ( I ˜ r \ I r ) | ≤ | I ˜ r \ I r | = 2 n (˜ r n − r n ) . The right-hand side is bounded as follows:2 n (˜ r n − r n ) ≤ n n (˜ r − r )˜ r n − ≤ n n h ˜ r n − ≤ n n − hr n − ≤ εr n if we require that(4.9) r ≥ r ≥ n n − hε − . Thus(4.10) w (˜ r, t + T ) − w ( r, t + T ) ≤ εr n . This and a trivial inequality w ( r, t ) ≤ w (˜ r, t ) imply that w ( r, t + T ) − w ( r, t ) ≥ w (˜ r, t + T ) − w (˜ r, t ) − ε r n ≥ b s ( r, t, T ) − εr n , where the second inequality follows from (4.8). This finishes the proof of Proposition 4.1. (cid:3) . Middle stage. Proof of Proposition 3.2
In this section we prove Proposition 3.2, the last remaining piece of the proof of Theorem1.2. The proof is based on Proposition 4.1 and the isoperimetric inequality (2.11) for subsetsof a cube.To facilitate understanding of the proof, we first give its simplified version assuming thatthe estimate (4.1) from Proposition 4.1 holds without the correction term − εr n . After thissimplification the estimate (4.1) boils down to the differential inequality(5.1) ddt w ( r, t ) ≥ s ( r, t ) ≥ λ min { w ( r, t ) , | I r | − w ( r, t ) } n − n where the second inequality is the isoperimetric inequality (2.11). This implies that w ( r, t ) ≥ φ ( t ) where φ ( t ) > ddt φ ( t ) = λ min { φ ( t ) , | I r | − φ ( t ) } n − n with the initial condition lim t → φ ( t ) = 0. The solution is given by φ ( t ) = ( at n , t ∈ [0 , b ] , | I r | − a (2 b − t ) n , t ∈ [ b, b ] , where a = ( λ /n ) n , and b = ( a | I r | ) /n = cr with c = 2 n − n nλ − . It reaches the value φ ( t ) = | I r | at t = 2 b = 2 cr , and the coefficient 2 c depends only on n . This proves the maintheorem under the above simplifying assumption.The actual proof of Proposition 3.2 is essentially a discrete version of the above argument.We apply Proposition 4.1 to T = βr where β ∈ (0 ,
1) is a carefully chosen constant (de-pending on the flow but not depending on r ). This yields a lower bound for w ( r, T k ) where T k = T + kβr , k = 1 , , . . . . It turns out that for a sufficiently small ε > b s ( r, t )dominates the correction term − εr n and hence the resulting bound for w ( r, T k ) is similar tothe formula for φ ( T k ). This implies the desired conclusion.Another technical issue is that the isoperimetric inequality (2.11) does not integrate wellover time intervals. This is handled in Lemma 5.1 below, where we prove a discrete analogueof the differential inequality (5.1).Now we are back to the formal proof. Recall that we have a fixed α defined by (3.1). Wenow choose a small constant β ∈ (0 , β < α . Second, we requirethat β is so small that the following holds. For all x ∈ [ α ,
1] and all δ ∈ [0 , β ](5.2) ( x + δ ) /n − x /n ≥ n x − nn δ. Such β exists since the function x x /n is smooth on [ α ,
1] and its derivative equals n x − nn .We fix α and β for the rest of the proof. Lemma 5.1.
There exist λ = λ ( n ) ∈ (0 , and r > such that for every r ≥ r and T = βr the following holds.1. For all t > and τ ∈ [ t, t + T ] , (5.3) w ( r, τ ) ≥ w ( r, t ) − α | I r | . . If t > satisfies (5.4) α | I r | ≤ w ( r, t ) ≤ (1 − α ) | I r | , then (5.5) w ( r, t + T ) ≥ w ( r, t ) + λT m ( t ) n − n where m ( t ) = min { w ( r, t ) , | I r | − w ( r, t ) } . Proof.
Fix a sufficiently small ε >
0, namely ε < min { α , λ αβ } , where λ = λ ( n ) is the isoperimetric constant from (2.11). By Proposition 4.1 there exists r > w ( r, τ ) − w ( r, t ) ≥ b s ( r, t, τ ) − εr n for any r ≥ r , T = βr , and τ ∈ [ t, t + T ]. Since b s ( r, t, τ ) ≥
0, this implies that w ( r, τ ) − w ( r, t ) ≥ − εr n > − α | I r | due to the choice of ε . This proves the first claim of the lemma.To prove the second one, define m = inf { m ( τ ) : τ ∈ [ t, t + T ] } and consider two cases: m < m ( t ) and m ≥ m ( t ). Case 1: m < m ( t ). Then m ( τ ) < m ( t ) for some τ ∈ [ t, t + T ]. The definition of m ( t )and (5.4) imply that(5.7) | w ( r, τ ) − w ( r, t ) | > α | I r | . The inequality (5.3) rules out the case w ( r, τ ) < w ( r, t ), hence w ( r, τ ) > w ( r, t ) + α | I r | . Combining this inequality with (5.3) applied to τ and t + T in place of t and τ , respectively,yields(5.8) w ( r, t + T ) ≥ w ( r, τ ) − α | I r | > w ( r, t ) + α | I r | . On the other hand, by the trivial estimate m ( t ) ≤ | I r | = (2 r ) n we have T m ( t ) n − n ≤ T (2 r ) n − = βr (2 r ) n − = β | I r | < α | I r | . This and (5.8) imply (5.5) for any λ ≤ Case 2: m ≥ m ( t ). By the isoperimetric inequality (2.11) for subsets of the cube, s ( r, τ ) ≥ λ m ( τ ) n − n ≥ λ m n − n ≥ λ m ( t ) n − n for all τ ∈ [ t, t + T ]. Hence(5.9) b s ( r, t, T ) ≥ λ T m ( t ) n − n . By (5.4), we have m ( t ) ≥ α | I r | = α (2 r ) n . Therefore(5.10) b s ( r, t, T ) ≥ λ T m ( t ) n − n = λ βrm ( t ) n − n ≥ λ βr (cid:0) α (2 r ) n (cid:1) n − n > λ αβr n > εr n , here the last inequality follows from the choice of ε . Inequalities (5.10), (5.6) and (5.9)imply that(5.11) w ( r, t + T ) − w ( r, t ) ≥ b s ( r, t, T ) − εr n ≥ b s ( r, t, T ) ≥ λ T m ( t ) n − n . The inequality (5.11) implies (5.5) for λ = λ .Combining the outcomes of the two cases, one sees that (5.5) holds for λ = min { , λ } . (cid:3) Now we are in a position to prove Proposition 3.2. The proof is a straightforward buttechnical implication of Lemma 5.1. Nothing beyond basic analysis is used.
Proof of Proposition 3.2.
Let r be such that the assertion of Lemma 5.1 holds. Fix r > r and define a function f : R + → [0 ,
1] by f ( t ) := w ( r, t ) | I r | = w ( r, t )(2 r ) n . We rewrite some of the previous results in terms of f . First, Lemma 3.1 turns into theinequality(5.12) f ( T ) ≥ α where T = r M . By the first statement of Lemma 5.1 we have(5.13) f ( τ ) ≥ f ( t ) − α if t ≤ τ ≤ t + βr. Finally, the second statement of Lemma 5.1 takes the form:(5.14) f ( t + βr ) ≥ f ( t ) + λβ min { f ( t ) , − f ( t ) } n − n provided that α ≤ f ( t ) ≤ − α . Here λ = λ ( n ) ∈ (0 ,
1] is the constant from Lemma 5.1, and we use this notation throughoutthe rest of the proof.In our new notation the statement of Proposition 3.2 turns into f ( t ) > − α for all t ≥ Ar where A is a constant depending only on n .Now consider a sequence { y k } ∞ k =0 defined by y k = f ( T + kβr ). The relations (5.12)–(5.14)imply the following properties of this sequence:(1) y ≥ α ;(2) if α ≤ y k ≤ then y k +1 ≥ y k + λβy n − n k ;(3) if ≤ y k ≤ − α then y k +1 ≥ y k + λβ (1 − y k ) n − n ;(4) if y k ≥ − α then y k +1 ≥ − α .It follows that { y k } increases as long as it stays below 1 − α , and if it gets above 1 − α thenafter that it is confined to the interval [1 − α, y k eventuallyattains a value greater than 1 − α , and estimate the index k for which this happens.If y k ≤ then by (1) and (2) we have y k ≥ y ≥ α and y k +1 ≥ y k + δ k where δ k = λβy n − n k .Hence y /nk +1 ≥ ( y k + δ k ) /n ≥ y /nk + 12 n y − nn k δ k = y /nk + λβ n . ere the second inequality follows from the choice of β (see (5.2)) and the fact that δ k ≤ β since λ ≤ y k ≤
1. By induction it follows that y /nk ≥ y /n + λβk n > λβk n as long as y , . . . , y k − ≤ . Hence there exists k ≤ nλβ such that y k ≥ .Now consider k ≥ k . Note that y k ≥ by (2)–(4). As long as y k ≤ − α , we have(5.15) y k +1 ≥ y k + δ k , where δ k = λβ (1 − y k ) n − n . We rewrite (5.15) as follows:(1 − y k +1 ) /n ≤ (1 − y k − δ k ) /n ≤ (1 − y k ) /n − n (1 − y k ) − nn δ k = (1 − y k ) /n − λβ n . Here the second inequality follows from the concavity of the function t t /n . By inductionit follows that (1 − y k ) /n ≤ (1 − y k ) /n − λβ n ( k − k ) ≤ − λβ n ( k − k )as long as y k , . . . , y k − ≤ − α . Hence there exists k ≤ k + nλβ ≤ nλβ such that y k ≥ − α .Then (3) and (4) imply that y k ≥ − α for all k ≥ k .This and (5.13) imply that f ( t ) ≥ − α for all t ≥ T + βrk . Since T + βrk ≤ T + nλ r ≤ ( nλ + 1) r , the statement of Proposition 3.2 holds for A = nλ + 1. (cid:3) Application to homogenization of the G-equation
In this section we prove a result about the homogenization limit of solutions to the G-equation with random drift. The proof of this result is a corollary of Theorem 1.2 combinedwith standard arguments of the homogenization theory. We give these arguments here forconvenience of the reader. We start with the notions needed to formulate our result.We investigate the asymptotic behavior as ε → ε . Namely, we consider the family of Hamilton-Jacobi equations: u εt + V tε (cid:16) xε , ω (cid:17) · Du ε = | Du ε | , t > , x ∈ R n , (6.1) u ε = u ( x ) , t = 0 , x ∈ R n , for the unknown u ε = u ε ( t, x, ω ), where u εt and Du ε are the derivatives of u ε with respectto t and x , respectively. Here ω is an elementary event (realization) in the sample space: ω ∈ Ω. We assume the sample space is a part of the probability triple (Ω , F , P ), where F isthe σ -algebra of measurable events, and P is the probability measure. The velocity V t : R n +1 × Ω → R n is a random field, a family of random variables parametrized by x and t . All random variablesare assumed Borel measurable.If V t is locally Lipschits, then, by, e.g. Exercise 3.9 in [2], we are guaranteed that theviscosity solutions of the G-equation (6.1) are unique in the space of bounded and uniformly ontinuous functions for every fixed ω . These solutions u ε ( t, x, ω ) of (6.1) are random func-tions in x and t . Our objective is to determine assumptions on V t ( x, ω ) that imply the lawof large numbers : u ε ( t, x, ω ) → ¯ u ( t, x ) with probability one as ε →
0, and characterize thedeterministic limit ¯ u ( t, x ) as a solution of another homogenized initial value problem. Inorder to determine this homogenized initial value problem, we will find a deterministic time-independent function ¯ H : R n → R + such that it is positively homogeneous of degree one,that is ¯ H ( λp ) = λ ¯ H ( p ) for all λ > p ∈ R n , and verify that ¯ u is the unique viscositysolution of the initial value problem¯ u t = ¯ H ( D ¯ u ) , x ∈ R n , t > , (6.2) ¯ u (0 , x ) = u ( x ) , x ∈ R n . The solutions of (6.1) have a control-representation formula (2.6). Similarly solutions of (6.2)are given by the Hopf-Lax formula [8, 10](6.3) ¯ u ( t, x ) = max { u ( y ) : ¯ T ( x − y ) ≤ t } , where ¯ T ( v ) = sup (cid:8) v · q : q ∈ R n , ¯ H ( q ) = 1 (cid:9) . The following two definitions are needed to state our assumptions on V t ( x, ω ). Definition 6.1.
We say that V t ( x, ω ) is space-time stationary if there is an action of R n +1 onΩ, denoted by y π y : Ω → Ω, y = ( x, t ) ∈ R n +1 , such that the action is measure-preserving:(6.4) P ( π y ( A )) = P ( A ) , ∀ A ∈ F , y ∈ R n +1 , and(6.5) V t ( x , π y ω ) = V t + t ( x + x, ω ) , ∀ x ∈ R n , t ∈ R , y = ( x, t ) ∈ R n +1 . Definition 6.2.
Define(6.6) G t + := σ { V s ( x, ω ) : s ≥ t, x ∈ R n } , G t − := σ { V s ( x, ω ) : s ≤ t, x ∈ R n } , where σ { . . . } denotes the σ -algebra on Ω generated by the given family of random variables.We say V t has finite range of time dependence if(6.7) ∃ℵ > G t + and G s − are independent when t − s ≥ ℵ . We state the result in two essentially equivalent ways.
Theorem 6.3.
Suppose that a random vector field V t : R n +1 × Ω → R n is time-space station-ary (6.4) – (6.5) , has finite range of time dependence (6.7) , V t ( · , ω ) is locally Lipschitz andincompressible for all t and ω , and has the following uniform bounds: (6.8) M := 1 + sup t,x,ω | V t ( x, ω ) | < ∞ , (6.9) ∆ := inf L> sup t,x,ω (cid:13)(cid:13)(cid:13)(cid:13) L n Z [0 ,L ] n V t ( x + y, ω ) dy (cid:13)(cid:13)(cid:13)(cid:13) < . Then there exists a convex body W ⊂ R n such that B − ∆ (0) ⊂ W ⊂ B M (0) and lim t →∞ d H ( t − R t ( ω ) , W ) = 0 for a.e. ω ∈ Ω , where R t ( ω ) is the reachable set from (0 , at time t (see Section 2) of theflow V t ( x, ω ) and d H denotes the Hausdorff distance. heorem 6.4. Let V t : R n +1 × Ω → R n be a random vector field satisfying the same assump-tions as in Theorem 6.3. Then there exists a positively one-homogeneous convex Hamiltonianfunction ¯ H : R n → [0 , ∞ ) with − ∆ ≤ ¯ H ( p ) / | p | ≤ M such that the following holds withprobability one: For every bounded uniformly continuous function u : R n → R one has (6.10) ∀ T > ∀ R > ε → sup t ∈ [0 ,T ] sup | x |≤ R | u ε ( t, x, ω ) − ¯ u ( t, x ) | = 0 , where u ε and ¯ u are the unique viscosity solutions of (6.1) and (6.2) , respectively. Remark 6.5.
Theorems 6.3 and 6.4 are also true if we request V t to be merely integerstationary. This means that (6.4)-(6.5) holds for y = ( x, t ) ∈ Z n +1 only. Here is an exampleof an integer stationary and finite range dependent flow V t ( x, ω ) that satisfies the conditionsof Theorem 6.3. Take any two deterministic incompressible vector fields V t ( x ) and V t ( x )with compact support in R n +1 . The incompressibility and compact support imply that(6.11) Z R n V it ( x ) dx = 0 , i = 1 , , for every t . Consider a family of Bernoulli trials, that is ζ jk ( ω ), j ∈ Z n , k ∈ Z are independentidentically distributed random variables such that ζ jk = 1 or ζ jk = 0 with probability 1 / V t ( x, ω ) = X j ∈ Z n ,k ∈ Z (cid:0) ζ jk ( ω ) V t + k ( x + j ) + (1 − ζ jk ( ω )) V t + k ( x + j ) (cid:1) . The identity (6.11) implies that this random field satisfies (6.9) with ∆ = 0.
Remark 6.6.
Using Theorem 1.2 and Corollary 1.3 we can prove the conclusions of The-orems 6.3 and 6.4 if, instead of finite range dependence and stationarity, we impose otherassumptions on V t . We are aware of two approaches. • If V t is periodic in x and random, statistically stationary and ergodic with respect to t , then the homogenization limit can be proven by an argument given in [9]. • If V t is periodic in t and random, statistically stationary and ergodic with respect to x , then the homogenization limit can be proven by an argument given in [13].Note that the level-set equation (6.1) is used as a model for turbulent combustion in theregime of thin flames [15, 14]. In this model, the level sets of u ε represent the flame surface,and V t is the velocity of the underlying fluid (assumed to be independent of u ε ). Spatial ortemporal periodicity is rarely observed in unsteady turbulent flows. Thus, in the context ofunsteady turbulent flows it is more relevant to assume the velocities are time-space stationaryand have finite range of time dependence.We prove Theorems 6.3 and 6.4 for a time-space stationary random vector field. Gener-alization to the integer stationary case is straightforward. We denote by R t ( x , t , ω ) thereachable set from ( x , t ) at time t + t of the flow V t ( x, ω ). Note that R t ( ω ) = R t (0 , , ω ).Observe that(6.12) R t ( x , t , ω ) ⊂ B Mt ( x ) ∀ t > , x ∈ R n , t ∈ R , ω ∈ Ω . Define Λ = − ∆ . Corollary 1.3 implies that there is a positive integer τ ∈ N such that(6.13) B t/ Λ ( x ) ⊂ R t ( x , t , ω ) ∀ t ≥ τ − , x ∈ R n , t ∈ R , ω ∈ Ω . ere we use (6.8), (6.9) and Remark 3.5 to ensure that τ is independent of ω . We assumethat τ > ℵ where ℵ is the range of time dependence from (6.7).The relation (6.13) implies that x ∈ R t ( x , t , ω ) for all t ≥ τ −
1. Therefore(6.14) R t ( x , t , ω ) ⊂ R t + t ( x , t , ω ) ∀ t ≥ τ − , t ≥ , x ∈ R n , t ∈ R , ω ∈ Ω . For x , v ∈ R n , t ∈ R and ω ∈ Ω, define the travel-time(6.15) τ ( x , t , v, ω ) = inf { t ∈ N : x + v ∈ R t ( x , t , ω ) } + τ . Set τ ( v, ω ) = τ (0 , , v, ω ). Note that for any N ∈ N the event { ω ∈ Ω : τ ( x , t , v, ω ) = N } is determined by the restriction of V t to the time interval [ t , t + N − τ ].By (6.12) and (6.13), the random variable τ ( v, ω ) grows linearly in v and moreover(6.16) | v | M ≤ τ ( x , t , v, ω ) ≤ Λ | v | + 2 τ for all x , t , v , ω . This estimate is the main ingredient of the first steps of the proof. Wealso need a number of technical estimates. By (6.14) we have(6.17) x + v ∈ R t ( x , t ) ∀ t ≥ τ ( x , t , v, ω ) − τ ( x , t , v, ω ) ≤ t + 2 τ if x + v ∈ R t ( x , t ) . For any x , x , v , v ∈ R n and t ∈ R we have(6.19) τ ( x , t , v , ω ) ≤ τ ( x , t + T, v , ω ) + 2 T ∀ T ≥ Λ | x − x | + Λ | v − v | + τ . Indeed, ( x , t + T ) is reachable from ( x , t ) by (6.13). Then the point x + v is reachablefrom ( x , t + T ) at time t = t + T + τ ( x , t + T, v , ω ) − τ . Then, by (6.13), x + v isreachable from ( x + v , t ) at any time t ≥ t + T −
1. Choosing t such that t − t is aninteger and t ≤ t + T yields (6.19).Our preliminary goal is to obtain the asymptotic shape of the reachable set. This is anal-ogous to “shape theorems” for the first-passage time in percolation theory, and we proceedwith similar arguments. Lemma 6.7.
There exists a positively -homogeneous convex function T : R n → R + satisfying (6.20) | v | M ≤ T ( v ) ≤ | v | − ∆ for all v ∈ R n and such that the following holds: i. For any v ∈ R n , x ∈ R n , t ∈ R , lim sup λ →∞ λ τ ( λx , λt , λv, ω ) = T ( v ) almost surely . ii. For any v ∈ R n , x ∈ R n , t ∈ R , λ τ ( λx , λt , λv, ω ) → T ( v ) n probability as λ → ∞ , that is (6.21) lim λ →∞ P (cid:26) ω : (cid:12)(cid:12)(cid:12)(cid:12) λ τ ( λx , λt , λv, ω ) − T ( v ) (cid:12)(cid:12)(cid:12)(cid:12) ≥ ε (cid:27) = 0 for every ε > .Proof. Fix x , v , v ∈ R n , t ∈ R , and define τ ( ω ) = τ ( x , t , v , ω ). By (6.17) and thedefinition of τ we have the following sub-additivity relation:(6.22) τ ( x , t , v + v , ω ) ≤ τ ( ω ) + τ ( x + v , t + τ ( ω ) , v , ω ) . The two terms in the right-hand side of (6.22) are independent random variables and theyhave the same distributions as τ ( v , · ) and τ ( v , · ), respectively. To show this, fix any N , N ∈ N and consider events A N = { ω : τ ( ω ) = N } , and B N ,N = { ω : τ ( x + v , t + N , v , ω ) = N } . Due to the space-time stationarity, their probabilities are equal to those of { τ ( v , · ) = N } and { τ ( v , · ) = N } , respectively. The event A N is determined by V t ( x, ω ) for t ≤ t + N − τ and B N ,N is determined by V t ( x, ω ) for t ≥ t + N . Since τ > ℵ , the finite range of timedependence implies that A N ,N and B N are independent. Thus(6.23) P ( { ω : τ ( ω ) = N and τ ( x + v , t + τ ( ω ) , v , ω ) = N } ) = P ( A N ∩ B N ,N )= P ( A N ) P ( B N ,N ) = P ( { τ ( v , · ) = N } ) P ( { τ ( v , · ) = N } ) . Summing over either N or N we obtain that τ ( ω ) and τ ( x + v , t + τ ( ω ) , v , ω ) have thesame distributions as τ ( v , · ) and τ ( v , · ), respectively; furthermore (6.23) shows that theyare independent.Therefore, from (6.22) we have(6.24) E ( τ ( v + v , · )) ≤ E ( τ ( v , · )) + E ( τ ( v , · )) . This implies that there exists a limit(6.25) T ( v ) := lim λ →∞ E ( τ ( λv, · )) λ = inf λ> E ( τ ( λv, · )) λ . The function T is 1-homogeneous by definition. By (6.24), T is sub-additive and henceconvex. The inequality (6.16) implies that | v | /M ≤ T ( v ) ≤ Λ | v | . Moreover, by Corollary1.3 for every a > − ∆ there is a constant C > τ ( v, ω ) ≤ a | v | + C for all v ∈ R n and ω ∈ Ω. Hence T ( v ) ≤ a | v | for all a > − ∆ and (6.20) follows.Fix v ∈ R n and arbitrary sequences { x k } ⊂ R n and { t k } ⊂ R , k ∈ N . For each k , definefinite sequences ξ k,m and t k,m , 1 ≤ m ≤ k , of random variables by induction as follows: ξ k,m ( ω ) = τ ( x k + ( m − v, t k,m ( ω ) , v, ω ) , where t k,m ( ω ) = t k + m − X i =1 ξ k,i ( ω ) , in particular t k, ( ω ) = t k . Note that for any N ∈ N the event { ω : t k,m ( ω ) = t k + N } isdetermined by the values V t ( x, ω ) for t ∈ [ t k , t k + N − τ ] only. As in the above discussion f the terms in (6.22), one sees that for each fixed k the random variables ξ k,m , 1 ≤ m ≤ k ,are independent and have the same distribution as τ ( v, · ). Since ξ k,m are uniformly bounded(see (6.16)), the strong law of the large numbers for triangular arrays applies to them, andwe obtain that(6.26) lim k →∞ k k X m =1 ξ k,m ( ω ) = E ( τ ( v, · )) , almost surely.As in (6.22) we have sub-additivity τ ( x k , t k , kv, ω ) ≤ k X m =1 ξ k,m ( ω )for all k ∈ N and ω ∈ Ω. This and (6.26) imply that(6.27) lim sup k →∞ τ ( x k , t k , kv, ω ) k ≤ E ( τ ( v, · )) , almost surely . Now we prove the two main assertions of the lemma. Fix x , v ∈ R n , t ∈ R , and ε > λ > T ( v ) ≤ E ( τ ( λ v, · )) λ ≤ (1 + ε ) T ( v ) . For λ ≥ λ let k ∈ N be such that kλ ≤ λ < ( k + 1) λ . We apply (6.19) to λx , kλ x , λv , kλ v , λt in place of x , x , v , v , t , respectively, with T = T + ( kλ − λ ) t where T = Λ λ | x | + Λ λ | v | + λ | t | + τ . This implies that τ ( λx , λt , λv, ω ) ≤ τ ( kλ x , kλ t + T , kλ v, ω ) + 2 T + 2 λ | t | where the last term comes from the estimate | kλ − λ | ≤ λ . Thereforelim sup λ →∞ τ ( λx , λt , λv, ω ) λ ≤ lim sup k →∞ τ ( kλ x , kλ t + T , kλ v, ω ) kλ . By (6.27) applied to x k = kλ x , t k = kλ t + T , and λ v in place of v , the right-hand sideis bounded by E ( τ ( λ v, · )) /λ almost surely. Thuslim sup λ →∞ τ ( λx , λt , λv, ω ) λ ≤ E ( τ ( λ v, · )) λ ≤ (1 + ε ) T ( v ) , almost surely . Since ε is arbitrary, it follows that(6.28) lim sup λ →∞ τ ( λx , λt , λv, ω ) λ ≤ T ( v ) , almost surely . By the space-time stationarity and (6.25),(6.29) E (cid:18) τ ( λx , λt , λv, · ) λ (cid:19) = E (cid:18) τ ( λv, · ) λ (cid:19) ≥ T ( v ) . Since τ ( λx , λt , λv, · ) /λ is bounded above by Λ | v | + τ for all λ ≥
1, (6.28), (6.29) andFatou’s lemma imply thatlim sup λ →∞ τ ( λx , λt , λv, ω ) λ = T ( v ) , almost surely , nd τ ( λx , λt , λv, · ) /λ converges to T ( v ) in probability. (cid:3) Definition 6.8.
Let T be the function constructed in Lemma 6.7. Define the effectivereachable set W t = (cid:8) v ∈ R n : T ( v ) ≤ t (cid:9) . Note that W t = t · W and W is a convex body satisfying B − ∆ (0) ⊂ W ⊂ B M (0). Weare going to show that the reachable set R t ( x , t , ω ) for large t is close to the set x + W t ina certain sense. We introduce the following quantity measuring the difference between thesesets. Definition 6.9.
For x ∈ R n , t ∈ R , t ≥ τ and ω ∈ Ω define ρ + ( x , t , t, ω ) = inf { ε > R t ( x , t , ω ) ⊂ x + (1 + ε ) W t } ,ρ − ( x , t , t, ω ) = inf { ε > x + (1 + ε ) − W t ⊂ R t ( x , t , ω ) } and ρ ( x , t , t, ω ) = max { ρ + ( x , t , t, ω ) , ρ − ( x , t , t, ω ) } . Note that the statement of Theorem 6.3 is equivalent to the property thatlim t →∞ ρ (0 , , t, ω ) = 0 , almost surely . Lemma 6.10.
For any fixed
R > , (6.30) lim t →∞ sup | x |≤ Rt ρ − ( x , , t, ω ) = 0 almost surelyand (6.31) lim t →∞ sup | x |≤ Rt ρ + ( x , , t, ω ) = 0 in probability,that is for any ε > , (6.32) P { ω : ∀ x ∈ B Rt (0) , R t ( x , , ω ) ⊂ x + (1 + ε ) W t } → as t → ∞ . Proof.
To prove (6.30), fix
R > ε > ε -nets { y i } Ni =1 in the ball B R (0) and { v j } Kj =1 in the effective 1-reachable set W . For every x ∈ B Rt (0) and v ∈ W t there exist i and j such that | x − ty i | < tε and | v − tv j | < tε . Assuming that t ≥ ε − τ ≥ − ε − τ , wesee from (6.19) that τ ( x , , v, ω ) ≤ τ ( ty i , tε, tv i , ω ) + 6Λ tε for all ω ∈ Ω. Hence sup | x |≤ Rt,v ∈ W t τ ( x , , v, ω ) ≤ max i,j τ ( ty i , tε, tv j , ω ) + 6Λ tε for all t ≥ ε − τ and ω ∈ Ω. By Lemma 6.7 (part 1)lim sup t →∞ max i,j t τ ( ty i , tε, tv j , ω ) = max j T ( v j ) ≤ , almost surely . Thus lim sup t →∞ sup | x |≤ Rt,v ∈ W t t τ ( x , , v, ω ) ≤ ε, almost surely . y (6.17) this implies that for every δ > s = s ( δ, ω ) > x + v ∈ R t (1+6Λ ε + δ ) ( x , t )for all t ≥ s , v ∈ W t and | x | ≤ Rt . Setting δ = Λ ε we obtain that ρ − ( x , , t (1 + 7Λ ε ) , ω ) ≤ ε for all t ≥ s = s (Λ ε, ω ) and | x | ≤ Rt . Thereforelim sup t →∞ sup | x |≤ R ′ t ρ − ( x , , t, ω ) ≤ ε, almost surelywhere R ′ = (1 + 7Λ ε ) − R . Since R and ε are arbitrary, (6.30) follows. To prove (6.31), fix R > ε > ( t ) = (cid:8) ω : ∃ x ∈ B Rt (0) , ρ + ( x , , t, ω ) > ε (cid:9) . Let δ = ε/
32Λ and choose δ -nets { y i } Ni =1 in B R (0) and { v j } Kj =1 in B M (0). Consider ω ∈ Ω ( t )where t ≥ δ − τ . By the definition of Ω ( t ) there exist x ∈ B Rt (0) and v ∈ R t ( x , , ω ) − x such that v / ∈ (1 + ε ) W t . By (6.12) we have v ∈ B Mt (0), hence there exist i and j such that | x − ty i | < δt and | v − tv j | < δt . These inequalities, (6.19), and (6.18) imply that(6.33) τ ( ty i , − δt, tv j , ω ) ≤ τ ( x , , v, ω ) + 6Λ δt ≤ t + 2 τ + 6Λ δt ≤ (1 + ε/ t. Since v / ∈ (1 + ε ) W t , we have T ( t − v ) ≥ ε . On the other hand, T ( t − v ) ≤ T ( v j ) + T ( t − v − v j ) ≤ T ( v j ) + Λ | t − v − v j | ≤ T ( v j ) + Λ δ ≤ T ( v j ) + ε/ T and (6.20). Therefore T ( v j ) ≥ ε/
2. Hence, by (6.33),1 t τ ( ty i , − δt, tv j , ω ) ≤ ε/ ≤ T ( v j ) − ε/ . Thus P (Ω ( t )) ≤ X i,j P (cid:26) ω : 1 t τ ( ty i , − δt, tv j , ω ) ≤ T ( v j ) − ε/ (cid:27) for all t ≥ δ − τ . By Lemma 6.7 (part 2), each summand in the right-hand side goes to 0 as t → ∞ . Hence P (Ω ( t )) → t → ∞ and (6.31) follows. (cid:3) Definition 6.11.
Define the support function of W (a.k.a. the effective Hamiltonian)¯ H ( p ) = sup { p · y | y ∈ W } . Since ¯ H ( p ) is the supremum of a family of linear functions of p , it is immediate that ¯ H isconvex in p , and positively homogeneous of degree one. Since B − ∆ (0) ⊂ W ⊂ B M ( t ), wehave (1 − ∆) | p | ≤ ¯ H ( p ) ≤ M | p | . Similarly, we define the support functions of reachable sets. Definition 6.12.
For p ∈ R n , x ∈ R n , t ∈ R and ω ∈ Ω define H t ( x , t , p, ω ) = sup { p · ( y − x ) | y ∈ R t ( x , t , ω ) } and H t ( p, ω ) = H t (0 , , p, ω ) . Due to the space-time stationarity, the random variable H t ( x , t , p, · ) has the same dis-tribution as H t ( p, · ). emma 6.13. For any p ∈ R n and R > , (6.34) lim sup t →∞ sup | x |≤ Rt H t ( x , , p, ω ) t ≤ ¯ H ( p ) , almost surely. Here is an outline of the proof of Lemma 6.13. First we adjust parameters in (6.34) to definea more manageable random variable h ( t, ω ), see (6.37) and (6.38) below. The advantagesof h ( t, ω ) over the original expression are its sub-additivity and independence properties,demonstrated in the course of the proof. With the new variable h ( t, ω ) the lemma is reducedto (6.39), which we then prove in four steps. In Step 1 we prove the sub-additivity (6.43).Unfortunately this sub-additivity is weaker than the classical one; we only have a bound for h ( qt, ω ) by a sum of h q ( t, ω ) where h q is another random variable parametrized by q ∈ N . Weovercome this difficulty by chaining random variables h q ( t, ω ) to h ( t, ω ) in Step 2. Namely,we show in Step 2 that one can control distributions of h q ( t, ω ) by distribution of h ( t, ω ),see (6.45). Step 3 is the key one. There we prove almost sure convergence for t ranging alonga geometric progression, see (6.51). We do this by analysis of the probability distribution of h ( t, ω ) using our stationarity and independence assumptions, sub-additivity of h ( t, ω ) and itsconvergence in probability (6.40). In our final Step 4 we show that the linear bound (6.36)on the growth of H t ( x , t , p, ω ) is sufficient to deduce the convergence for all t → ∞ . Proof of Lemma 6.13.
We begin with several preliminary observations. By scaling it is suf-ficient to consider p ∈ R n with | p | = 1. We may also assume that R ≥ M . We fix such p and R for the rest of the proof. Since R t ( x , t , ω ) ⊂ B Mt ( x ), we have(6.35) H t ( x , t , p, ω ) ≤ M t.
Moreover,(6.36) H t + t ( x , t , p, ω ) ≤ H t ( x , t , p, ω ) + M t for all t , t ≥
0, since R t + t ( x , t ) is contained in the ( M t )-neighborhood of R t ( x , t ).For x ∈ R n , t ∈ R , t ≥ τ , ω ∈ Ω, define(6.37) h ( x , t , t, ω ) = sup | x − x |≤ Rt H t − τ ( x, t , p, ω ) + M τ and, for brevity,(6.38) h ( t, ω ) = h (0 , , t, ω ) . For every x ∈ B Rt (0) we have H t ( x , , p, ω ) ≤ h ( t, ω ) by (6.36) applied to t = t − τ and t = τ . Thus, in order to prove the Lemma it suffices to show that(6.39) lim sup t →∞ h ( t, ω ) t ≤ ¯ H ( p ) . Let us now reformulate the convergence in probability from Lemma 6.10 in terms of h ( t, ω ).We claim that, for every ε > P (cid:26) ω : h ( t, ω ) t > ¯ H ( p ) + ε (cid:27) → t → ∞ . Indeed, by (6.32) in Lemma 6.10 we have(6.41) P { ω : ∀ x ∈ B Rt (0) , R t ( x , , ω ) − x ⊂ (1 + ε ) W t } → t → ∞ . or every ω satisfying the relation R t ( x , , ω ) − x ⊂ (1 + ε ) W t in (6.41), we have H t ( x , , p, ω ) ≤ sup { p · y | y ∈ (1 + ε ) W t } = (1 + ε ) t ¯ H ( p ) . Therefore, we can conclude from (6.41) that P (cid:26) ω : ∀ x ∈ B Rt (0) , H t ( x , , p, ω ) t ≤ (1 + ε ) ¯ H ( p ) (cid:27) → t → ∞ , for every ε >
0, and (6.40) follows.In order to state sub-additivity properties of h ( t, ω ) we need one more definition. Fix q ∈ N and define h q ( x , t , t, ω ) = sup | x − x |≤ qRt H t − τ ( x, t , p, ω ) + M τ and h q ( t, ω ) = h q (0 , , t, ω )for x ∈ R n , t ∈ R , t ≥ τ , ω ∈ Ω. (The reader may notice that for q = 1 one has h q ( t, ω ) = h ( t, ω ) but we do not need this fact in the proof). Observe that(6.42) h q ( x , t , t, ω ) ≤ M t by (6.35). We are now ready for our four steps.
Step 1 . Sub-additivity of h ( t, ω ) . We show here that for every q ∈ N , t ≥ τ , ω ∈ Ω,(6.43) h ( qt, ω ) ≤ q − X k =0 h q (0 , kt, t, ω ) . Indeed, let γ : [0 , qt − τ ] → R n be an admissible path for V t ( x, ω ) with γ (0) ∈ B Rt (0). Toprove (6.43), it suffices to verify that(6.44) ( γ ( qt − τ ) − γ (0)) · p ≤ q − X k =0 h q (0 , kt, t, ω ) − M τ for every such path γ . Observe that γ ( kt ) ∈ B qRt (0) for k = 0 , . . . , q − γ (0) ∈ B Rt (0)and | ˙ γ | ≤ M ≤ R . Hence( γ (( k + 1) t − τ ) − γ ( kt )) · p ≤ H t − τ ( γ ( kt ) , kt, p, ω ) ≤ h q (0 , kt, t, ω ) − M τ for each k = 0 , , . . . , q −
1. We also have( γ ( kt ) − γ ( kt − τ )) · p ≤ | γ ( kt ) − γ ( kt − τ ) | ≤ M τ for each k = 1 , . . . , q − q − Step 2 . Chaining of h q ( t, ω ) . The goal of this step is to show that there exists N = N ( q, n ) ∈ N such that(6.45) P { ω : h q ( t, ω ) > α } ≤ N · P { ω : h ( t, ω ) > α } for all α ∈ R , t ≥ τ , ω ∈ Ω.To prove this, observe that a ball of radius qRt can be covered by N balls of radius Rt : B qRt (0) ⊂ N [ i =1 B Rt ( z i ) or some z , . . . , z N , where N is determined by q and n . Therefore h q ( t, ω ) ≤ max ≤ i ≤ N h ( z i , , t, ω ) , hence P { ω : h q ( t, ω ) > α } ≤ N X i =1 P { ω : h ( z i , , t, ω ) > α } . Due to the space-time stationarity, each summand in the last sum equals P { ω : h ( t, ω ) > α } and the inequality (6.45) follows. Step 3 . Convergence along a geometric progression.
As we have mentioned earlier, this isthe key step. Recall that our goal is to prove (6.39). Here we prove that the same inequalitywith a small error term holds for t ranging along a geometric progression with commonratio q , see (6.51) below.Fix ε > q ∈ N , and let N = N ( q, n ) from Step 2. Define f ( t, ω ) = h ( t, ω ) t − ¯ H ( p ) − ε and f k ( t, ω ) = h q (0 , kt, t, ω ) t − ¯ H ( p ) − ε. for all t ≥ τ , ω ∈ Ω, k ∈ { , . . . , q } . Note that f k ( t, ω ) ≤ M by (6.42).With this notation, (6.43) takes the form(6.46) f ( qt, ω ) ≤ q q − X k =0 f k ( t, ω ) . The inequality (6.45) along with the space-time stationarity imply that(6.47) P { f k ( t, ω ) > α } ≤ N · P { f ( t, ω ) > α } for all α ∈ R .Fix a positive δ < q N . By (6.40), P { ω : f ( t, ω ) > } → t → ∞ . Hence there exists t ≥ τ such that(6.48) P { ω : f ( t, ω ) > } < δ ∀ t ≥ t . Define(6.49) ∆( t ) = P (cid:26) ω : f ( t, ω ) > Mq (cid:27) for all t ≥ τ . We are going to estimate ∆( qt ) in terms of ∆( t ) using the above inequalities.Assume that t ≥ t where t is the same as in (6.48). The bound f k ( t, ω ) ≤ M and (6.46)imply the following property: For every ω ∈ Ω such that f ( qt, ω ) > Mq , at least two of theterms f k ( t, q ) must be positive and at least one of them must be greater than Mq . Therefore(6.50) ∆( qt ) ≤ X i = j P (cid:26) ω : f i ( t, ω ) > Mq and f j ( t, ω ) > (cid:27) . bserve that the random variables f i ( t, · ) and f j ( t, · ) are independent if i = j . This followsfrom the finite range time dependence and the fact that f k ( t, ω ) is determined by the re-striction of the flow to the time interval [ kt, ( k + 1) t − τ ]. Hence (6.50) can be rewrittenas ∆( qt ) ≤ X i = j P { ω : f i ( t, ω ) > M/q } · P { f j ( t, ω ) > } . This and (6.47), (6.48), (6.49) imply that∆( qt ) ≤ X i = j N ∆( t ) · N δ = q ( q − N δ ∆( t ) ≤ ∆( t )2where the last inequality follows from the choice of δ .By induction it follows that ∆( q m t ) ≤ − m for all t ≥ t and m ∈ N . By the Borel-CantelliLemma and (6.49), this implies that for every t > m →∞ f ( q m t, ω ) ≤ Mq for a.e. ω ∈ Ω. Substituting the definition of f yields that(6.51) lim sup m →∞ h ( q m t, ω ) q m t ≤ ¯ H ( p ) + ε + Mq for a.e. ω ∈ Ω. Step 4 . Convergence for all t . To finish the proof, choose a partition 1 = t ≤ t ≤ · · · ≤ t l = q of [1 , q ] such that t i +1 < (1 + ε ) t i for all i < l . For every t ≥ q there exist positiveintegers m ∈ N and i < l such that q m t i ≤ t < q m t i +1 < q m t i + εt. These inequalities and (6.36) imply that h ( t, ω ) ≤ h ( q m t i , ω ) + M εt, and hencelim sup t →∞ h ( t, ω ) t ≤ lim sup m →∞ max ≤ i
For any fixed
R > t →∞ sup | x |≤ Rt ρ + ( x , , t, ω ) = 0 almost surely . roof. Fix
R > ε ∈ (0 , W is a compact convex set, we have W = { x ∈ R n : x · p ≤ ¯ H ( p ) , ∀ p ∈ R n } . Furthermore there is a finite collection of vectors p , . . . , p N ∈ R n with | p i | = 1 such that f W := { x ∈ R n : x · p i ≤ ¯ H ( p i ) , ∀ i } ⊂ (1 + ε ) W . By Lemma 6.13, for almost every ω ∈ Ω there exists t ω > t > t ω and x ∈ B Rt (0), ( x − x ) · p i ≤ (1 + ε ) t ¯ H ( p i ) , ∀ x ∈ R t ( x , , ω ) , ∀ i. This implies that R t ( x , , ω ) − x ⊂ (1 + ε ) t f W ⊂ (1 + ε ) W t and therefore ρ + ( x , , t, ω ) < (1 + ε ) − < ε . Since ε is arbitrary, (6.52) follows. (cid:3) Proof of Theorems 6.3 and 6.4.
Theorem 6.3 follows by setting W = W and applying(6.30) and (6.52).To prove Theorem 6.4 we recall the control representation (2.6) for the solution of theG-equations. For x ∈ R n , t > ω ∈ Ω define R − t ( x, ω ) = { y ∈ R n : x ∈ R t ( y, , ω ) } . The control representation for the solution of (6.1) and (6.2) have the form u ε ( t, x, ω ) = sup { u ( y ) : y ∈ ε R − t/ε ( x/ε, ω ) } and ¯ u ( t, x ) = sup { u ( y ) : y ∈ x − W t } . Let δ > h >
0, and
R >
0. From (6.30) and (6.52) we see that for almost every ω ∈ Ωthere exists ε = ε ( δ, R, h, ω ) > | x | ≤ R , t ≥ h , and ε ≤ ε we have { x − W t (1 − δ ) } ⊂ ε R − t/ε ( x/ε, ω ) ⊂ { x − W t (1+ δ ) } , Therefore(6.53) ¯ u ( t (1 − δ ) , x ) ≤ u ε ( t, x, ω ) ≤ ¯ u ( t (1 + δ ) , x ) . Since δ > u ( t, x ) is uniformly continuous, (6.53) implies that u ε → ¯ u uniformly on compact sets in (0 , ∞ ) × R n . To obtain the locally uniform convergence downto time t = 0, we need the uniform L ∞ bound on V t and uniform continuity of u ( x ). Observethat sup t ∈ [0 ,h ] | u ε ( t, x, ω ) − ¯ u ( t, x ) | ≤ sup t ∈ [0 ,h ] | u ε ( t, x, ω ) − u ( x ) | + sup t ∈ [0 ,h ] | ¯ u ( t, x ) − u ( x ) | . For any y ∈ ε R − t/ε ( x/ε, , ω ) we have | y − x | ≤ M t . Thus the first term on the right isbounded by(6.54) sup t ∈ [0 ,h ] | u ε ( t, x, ω ) − u ( x ) | ≤ sup y ∈ R n | y − x |≤ Mh | u ( y ) − u ( x ) | ≤ φ ( M h ) , here φ is the modulus of continuity of u ( x ). This and a similar bound on | ¯ u ( t, x ) − u ( x ) | implies that(6.55) lim h → lim sup ε → sup x ∈ R n t ∈ [0 ,h ] | u ε ( t, x, ω ) − ¯ u ( t, x ) | = 0 . Combining (6.53) and (6.55), we conclude that (6.10) holds with probability one. (cid:3)
Appendix A. Functions of bounded variation
We collect here needed facts about functions of bounded variation (BV functions) in R n , n ≥
2. We followed [1] and [12].
Definition A.1 (Proposition 3.6 and Definition 3.4 in [1]) . Let Ω ⊂ R n be an open set and u ∈ L (Ω). The variation of u in Ω, denoted by Var( u, Ω), isVar( u, Ω) = sup (cid:26)Z Ω u div φ : φ ∈ [ C c (Ω)] n , k φ k L ∞ ≤ (cid:27) . Here and below [ C c (Ω)] n denotes the set of all compactly supported C functions from Ω to R n .The space BV (Ω) consists of all functions u ∈ L (Ω) with Var( u, Ω) < ∞ . It is equippedwith the norm k u k BV = Z Ω | u | dx + Var( u, Ω) . Remark.
The distributional derivative Du of a BV-function u is a (vector-valued) finiteRadon measure, and Var( u, Ω) = | Du | (Ω). We occasionally writeVar( u, Ω) = Z Ω |∇ u | , where the right-hand side is understood in the sense of distributions. Definition A.2 (Definition 3.35 in [1]) . The perimeter P ( E, Ω) of a measurable set E ⊂ R n in an open set Ω ⊂ R n is defined by P ( E, Ω) = Var( χ E , Ω) = sup (cid:26)Z E div φ : φ ∈ [ C c (Ω)] n , k φ k L ∞ ≤ (cid:27) . We denote P ( E ) = P ( E, R n ).In all cases of interest in this paper the set E is bounded. Definition A.3 (Reduced boundary, Definition 3.54 in [1]) . Let E ⊂ R n be a set of finiteperimeter. The reduced boundary ∂ ∗ E of E is the collection of points x ∈ supp( | Dχ E | ) suchthat the limit(A.1) ν E ( x ) = lim ρ → R B ρ ( x ) ∇ χ E R B ρ ( x ) |∇ χ E | exists in R n and satisfies | ν E ( x ) | = 1. The integrals here are understood in the sense ofdistributions. The function ν E : ∂ ∗ E → S n − is called the generalized inner normal to E . heorem A.4 (De Giorgi Theorem, Theorem 15.9 in [12]) . If E ∈ R n is a set of finiteperimeter, then the reduced boundary ∂ ∗ E is H n − -rectifiable and P ( E, Ω) = H n − (Ω ∩ ∂ ∗ E ) for every open set Ω ⊂ R n . Recall that I r = [ − r, r ] n is a cube with edge length 2 r and I ◦ r denotes its interior. Theorem A.5 (Relative isopertimetric inequality in the cube) . If E is a set of finite perime-ter in R n , then for every r > | E ∩ I r | , | I r \ E | ) n − n ≤ CP ( E, I ◦ r ) = C H n − ( ∂ ∗ E ∩ I ◦ r ) , where C is a constant depending on n only.Proof. This inequality is standard but we could not find exactly this formulation in theliterature. For the sake of completeness we include a proof here.Every u ∈ BV ( I ◦ r ) satisfies the following Sobolev inequality (see e.g. Remark 3.50 in [12]):there is a constant C = C ( n ) such that(A.3) (cid:18)Z I r | u − u | nn − (cid:19) n − n ≤ C Var( u, I ◦ r ) , where u denotes the average of u over I r : u = 1 | I r | Z I r u. The fact that C does not depend on r follows from a scaling argument.Let u = χ E , then u = | E ∩ I r || I r | and 1 − u = | I r \ E || I r | , hence Z I t | u − u | nn − dx = (cid:18) | I r \ E || I r | (cid:19) nn − | E ∩ I r | + (cid:18) | E ∩ I r || I r | (cid:19) nn − | I r \ E | . Therefore (cid:18)Z I r | u − u | nn − dx (cid:19) n − n ≥ | I r | (cid:16) | E ∩ I r | nn − + | I r \ E | nn − (cid:17) n − n min ( | E ∩ I r | , | I r \ E | ) n − n ≥
12 min ( | E ∩ I r | , | I r \ E | ) n − n . This and the Sobolev inequality (A.3) implies the inequality in (A.2). The equality in (A.2)holds due to the De Giorgi Theorem A.4. (cid:3)
Corollary A.6. If E is a set of finite perimeter in R n , then for every r > | E ∩ I r | , | I r \ E | ) ≤ CrP ( E, I ◦ r ) = Cr H n − ( ∂ ∗ E ∩ I ◦ t ) where C is a constant depending only on n .Proof. The inequality follows immediately from (A.2) and the trivial estimatemin ( | E ∩ I r | , | I r \ E | ) ≤ | I r | = 2 n r n . (See also [1, Remark 3.45] for a different proof.) (cid:3) heorem A.7 (Federer Coarea Formula, Theorem 2.93 in [1]) . Let f : R n → R be a Lipschitzfunction and E ⊂ R n an H k -rectifiable set. Then the function t → H k − ( E ∩ f − ( t )) isLebesgue measurable, E ∩ f − ( t ) is H k − -rectifiable for almost every t ∈ R , and Z E |∇ τ f ( x ) | d H k ( x ) = Z ∞ t =0 H k − ( E ∩ f − ( t )) dt where ∇ τ f ( x ) is the component of ∇ f ( x ) tangential to E . In the next theorem we use the following notation. For t ∈ R , we denote by Σ t thehyperplane Σ t := { x ∈ R n : x = t } . For a set E ⊂ R n , we denote by E t the intersection(“slice”) E t := E ∩ Σ t . Corollary A.8 (Coarea inequality) . Let E ⊂ R n be a set with finite perimeter. Then ∂ ∗ E ∩ ∂I r is H n − -rectifiable for almost every r , and (A.4) H n − ( ∂ ∗ E ) ≥ Z ∞ H n − ( ∂ ∗ E ∩ ∂I r ) dr. Proof.
By the De Giorgi Theorem A.4 the reduced boundary ∂ ∗ E is H n − -rectifiable. Weobtain the inequality in (A.4) by applying Theorem A.7 to ∂ ∗ E in place of E with k = n − f ( x ) = k x k l ∞ ( R n ) , and using the fact that |∇ τ f ( x ) | ≤ (cid:3) Theorem A.9 (Boundary slicing theorem, Theorem 18.11 in [12]) . If E is a set of finiteperimeter in R n , then for almost every t ∈ R the slice E t = E ∩ Σ t is a set of finite perimeterin the hyperplane Σ t ∼ = R n − and H n − ( ∂ ∗ ( E t )∆( ∂ ∗ E ) t ) = 0 , where ∆ denotes symmetric difference of two sets and ∂ ∗ ( E t ) is the ( n − -dimensionalreduced boundary of E t in Σ t . References [1] L. Ambrosio, N. Fusco, D. Pallara, Functions of bounded variation and free discontinuity prob-lems, Oxford University Press, 2000.[2] M. Bardi, I. Capuzzo-Dolcetta,
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Dmitri Burago: The Pennsylvania State University, Department of Mathematics, Univer-sity Park, PA 16802, USA
E-mail address : [email protected] Sergei Ivanov: St. Petersburg Department of Steklov Mathematical Institute, RussianAcademy of Sciences, Fontanka 27, St.Petersburg 191023, Russia
E-mail address : [email protected] Alexei Novikov: The Pennsylvania State University, Department of Mathematics, Uni-versity Park, PA 16802, USA
E-mail address : [email protected]@math.psu.edu